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Theorem mptfcl 26675
Description: Interpret range of a maps-to notation as a constraint on the definition. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
mptfcl  |-  ( ( t  e.  A  |->  B ) : A --> C  -> 
( t  e.  A  ->  B  e.  C ) )
Distinct variable groups:    t, A    t, C
Allowed substitution hint:    B( t)

Proof of Theorem mptfcl
StepHypRef Expression
1 eqid 2412 . . 3  |-  ( t  e.  A  |->  B )  =  ( t  e.  A  |->  B )
21fmpt 5857 . 2  |-  ( A. t  e.  A  B  e.  C  <->  ( t  e.  A  |->  B ) : A --> C )
3 rsp 2734 . 2  |-  ( A. t  e.  A  B  e.  C  ->  ( t  e.  A  ->  B  e.  C ) )
42, 3sylbir 205 1  |-  ( ( t  e.  A  |->  B ) : A --> C  -> 
( t  e.  A  ->  B  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   A.wral 2674    e. cmpt 4234   -->wf 5417
This theorem is referenced by:  mzpsubmpt  26698  eq0rabdioph  26733  eqrabdioph  26734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429
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